"Geometry of the Quintic" is available for free at my website.
I covered Klein's "Lectures on the Icosahedron" in a modern way in my doctoral thesis:
Elliptic Curves and Icosahedral Galois Representations, Stanford University (1999) http://www.math.purdue.edu/~egoins/notes/thesis.pdf
A much shorter and more direct exposition is my publication in IMRN:
Icosahedral $\mathbb Q$-Curve Extensions, Mathematical Research Letters 10, 205–217 (2003) http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2003/0010/0002/MRL-2003-0010-0002-00019947.pdf
Chapter 5 of McKean and Moll's "Elliptic Curves" explores the circle of ideas around Ikosaeder.I'm not sure if you'd consider this sufficiently "modern" - it's certainly a contemporary book but it doesn't use, say, scheme-theoretic language.
In Glimpses of algebra and geometry by Gabor Toth, chapter 25 is devoted to Klein's main result.
There is a (german) new edition of Klein's "Vorlesungen über das Ikosaeder ..." by Peter Slodowy (1993) with a large (about 80 pages) section of comments and remarks about new developments.
There is the outrageously expensive "Geometry of the quintic" by Jerry Shurman, which discusses both Klein and Doyle-McMullen approaches (and then some more).
In my PH Thesis work http://systembit.es/schwarz.htm
In my papier I have asociatted a Riemann Surface to each Schwarz function triangle. After that ,I´ve got genus and geometric density of spherical tesselation expresions in a new method. Then I see Their Poincare Groups ( of each above Riemann Surfaces) as index normal finite subgroup of Г(2) (Thanks to Modular Function Lambda). Then I calculate signature of these fuchisian Groups. Finally I see there are only nine ( of above Riemann Surfaces) more Dihedrical cases.
I think my idea is a new interpretation of Schwarz triangles , different one to the Famous Schwarz Classification based on 14 Schwarz triangles +Dihedrical cases.
Alfonso García firstname.lastname@example.org
You could also take a look at Section 1.6 of Finite Mobius Groups, Immersion of Spheres, and Moduli, by Gabor Toth.